Multiple edges in M. Kontsevich's graph complexes and computations of the dimensions and Euler characteristics

Multi-directed Graph Complexes and Quasi-isomorphisms Between Them II: Sourced Graphs

“…Kontsevich's graph complex fGC n is originaly defined by M. Kontsevich in [5], [6], cf. [12] and [15]. The complex defined in some of those papers is the dual of the one we define here, the differential being the vertex splitting, but it does not change the homology.…”

The heptagon-wheel cocycle in the Kontsevich graph complex

“…We finally note that the numbers of nonzero graphs with a specified number of vertices and edges (and N(v) > 2), which we list in Table 3, all coincide with the respective entries in Table II in the paper [17].…”

“…The graphs which constitute γ 7 are drawn on pp. [13][14][15][16][17][18][19] in Appendix A. The code in Sage programming language, allowing one to calculate the differential for a given graph γ and ordering E(γ) on the set of its edges, is contained in Appendix B; the same code can be run to calculate the dimension of graph cohomology groups.…”

Euler characteristics for spaces of string links and the modular envelope of $\mathcal{L}_{\infty}$

“…Direct computations of the supercharacter of the symmetric group action on Mod(L ∞ ) and Mod Det (L ∞ ) using graph complexes have previously been much less tractable than we find here. Such were first attempted in case of arity zero, where these correspond to the well known Kontsevich graph-complexes associated to the commutative operad [24], by Willwacher andŽivković in [37]. The original paper of Getzler and Kapranov [14], where the notion of a modular operad is introduced, gives a universal method to compute the supercharacter of a Feynman transform of any modular operad, but explicit calculations were done in [14] only for the associative operad.…”